External Links

Table of Contents

Berkey et al. (1995)

The Methods and Data

Berkey et al. (1995) describe the meta-analytic random- and mixed-effects models and provide the equation for the empirical Bayes estimator for the amount of (residual) heterogeneity (p. 398). The models and methods are illustrated with the BCG vaccine dataset (Colditz et al., 1994). The data are provided in Table 1 in the article and can be loaded with:

The metafor package calculates the sampling variances of the log risk ratios as described by Berkey et al. (1995) on page 399 (right under the 2×2 table). However, note that Berkey et al. (1995) actually use "smoothed" estimates of the sampling variances for the actual analysis (to reduce the correlation between the log risk ratios and corresponding sampling variances). We can compute these estimates with:

These results match exactly what Berkey et al. (1995) report on page 408: The amount of heterogeneity ("between-trial variance") is estimated to be $\hat{\tau}^2 = 0.268$ and the pooled estimate is $\hat{\mu} = -0.5429$ with a standard error of $SE[\hat{\mu}] = 0.1842$.

Mixed-Effects Model

A mixed-effects model with absolute latitude as moderators (centered at 33.46 degrees) can be fitted with:

Again, the results match the findings from Berkey et al. (1995): The residual amount of heterogeneity is now $\hat{\tau}^2 = 0.157$ and the estimated model is $log(RR) = -0.6303 - 0.0268 (x - 33.46)$, where $x$ is the distance from the equator (in degrees latitude). The standard errors of the model coefficients are $SE[b_0] = 0.1591$ and $SE[b_1] = 0.0110$.

The amount of variance (heterogeneity) accounted for by the absolute latitude moderator is provided in the output above. It can also be obtained with: